40 
small a value we attribute to ¢, the series must always diverge 
after a certain number of terms.”* 
The reasoning by which a finite sum is determined for s, 
when ¢ = 1, is as follows: 
sdt 
>= l.dt—1.2¢dt+ 1.2.3@dt — &c. (7) 
os = =t—1.+1.20 — &c. (8) 
—t— st (9) 
OF = = (1 — s)dt — tds, 
or, 
Be ak pees 
dt "a oe 
and from this is found, for s, the definite integral 
1i¢ 
es ) edt; 
rf 0 
from which it is inferred that “if ¢ = 1, or the above integral 
be taken from ¢ = 0 to ¢ = 1, we have the expression for the 
value of the series 
1 — 1.2 + 1.2.3 — &e.” 
Now several objections lie against the preceding reasoning : 
in the first place it is assumed, in the final step, that s vanishes, 
for t= 0, notwithstanding that ‘* however small a value we 
attribute to ¢ the series must always diverge,” and thus at 
length furnish terms infinitely great: and in the next place it 
is assumed—and the assumption is somewhat similar to that 
* If, however, ¢ be indefinitely near to zero, the ‘‘certain number of 
terms” adverted to in the text, will be indefinitely great; that is, the diver- 
gency will be indefinitely postponed: the series therefore cannot be consi- 
dered as divergent up to the limit t=0; yet, as the statement in the text 
seems to imply this, I have considered it to be comprehended in the hypo- 
thesis; although, as I have shewn, the point is of no moment in the matter 
under discussion. 
